If a*b means a is added to b,
a $ b means b is subtracted from a.
a # b means a is multiplied by b.
a & b means a is divided by b,
then answer the following questions :
If `a=(x)/(2)+(3y)/(4)" and "b= (-x)/(2)+(6y)/(4)`, then find a $ b * a.

Options

A. `(1)/(2)x`
B. `(3)/(2)x`
C. `(3)/(4)y`
D. `(3)/(8)y`

Answer

To solve the problem, we need to follow the operations defined for \( a \) and \( b \) and then compute \( a \$ b * a \).

### Step-by-Step Solution:

1. **Define \( a \) and \( b \)**:
\[ a = \frac{x}{2} + \frac{3y}{4} \]
\[ b = -\frac{x}{2} + \frac{6y}{4} \]

2. **Calculate \( a \$ b \)**:
The operation \( a \$ b \) means \( a \) minus \( b \):
\[ a \$ b = a - b \]
Substitute the values of \( a \) and \( b \):
\[ a - b = \left( \frac{x}{2} + \frac{3y}{4} \right) - \left( -\frac{x}{2} + \frac{6y}{4} \right) \]

3. **Simplify \( a - b \)**:
Distributing the negative sign:
\[ a - b = \frac{x}{2} + \frac{3y}{4} + \frac{x}{2} - \frac{6y}{4} \]
Combine like terms:
\[ = \left( \frac{x}{2} + \frac{x}{2} \right) + \left( \frac{3y}{4} - \frac{6y}{4} \right) \]
\[ = \frac{2x}{2} - \frac{3y}{4} \]
\[ = x - \frac{3y}{4} \]

4. **Calculate \( a \$ b * a \)**:
Now we need to multiply \( a \$ b \) by \( a \):
\[ a \$ b * a = (x - \frac{3y}{4}) * a \]
Substitute \( a \):
\[ = (x - \frac{3y}{4}) \left( \frac{x}{2} + \frac{3y}{4} \right) \]

5. **Expand the expression**:
Using the distributive property:
\[ = x \left( \frac{x}{2} + \frac{3y}{4} \right) - \frac{3y}{4} \left( \frac{x}{2} + \frac{3y}{4} \right) \]
\[ = \frac{x^2}{2} + \frac{3xy}{4} - \left( \frac{3xy}{8} + \frac{9y^2}{16} \right) \]

6. **Combine like terms**:
To combine the terms, we need a common denominator:
\[ = \frac{x^2}{2} + \left( \frac{3xy}{4} - \frac{3xy}{8} \right) - \frac{9y^2}{16} \]
The common denominator for \( \frac{3xy}{4} \) and \( \frac{3xy}{8} \) is 8:
\[ = \frac{x^2}{2} + \left( \frac{6xy}{8} - \frac{3xy}{8} \right) - \frac{9y^2}{16} \]
\[ = \frac{x^2}{2} + \frac{3xy}{8} - \frac{9y^2}{16} \]

7. **Final expression**:
To express \( \frac{x^2}{2} \) with a common denominator of 16:
\[ = \frac{8x^2}{16} + \frac{6xy}{16} - \frac{9y^2}{16} \]
\[ = \frac{8x^2 + 6xy - 9y^2}{16} \]

Thus, the final answer is:
\[ \frac{8x^2 + 6xy - 9y^2}{16} \]